心臓病診療プラクティス 16. 心不全を癒す

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/63B506EB85D6875966CE63BA0336BC2F/S0025557200190515a.pdf/div-class-title-span-class-italic-lie-groups-and-lie-algebras-chapters-1-3-span-by-n-bourbaki-pp-450-dm-98-1989-isbn-3-540-50218-1-springer-div.pdf

Lie groups and Lie algebras (chapters 1-3 ), by N. Bourbaki. Pp 450. DM 98. 1989. ISBN 3-540-50218-1 (Springer)

Table of Contents:* Parameters * Local transfer * Global stabilization * The standard model * A study of critical cases * The local classification * Local nontempered representations * The global classification * Inner forms * Bibliography * Index

/pdf/the-endoscopic-classification-of-representations-4x2ckk53u0.pdf

The Endoscopic Classification of Representations

Data mining and machine learning methods for sustainable smart cities traffic classification: a survey

LetG be a reductive p-adic group. We give a new construction of small-depth “epipelagic” supercuspidal representations ofG(k), using stable orbits in Geometric Invariant Theory (GIT). In contrast to previously known methods, this construction works without any restrictions on the residue characteristic p. We construct appropriate Langlands parameters for these representations, with some restrictions on p. The GIT arises from graded Moy-Prasad filtrations, which we show are isomorphic to the graded Lie algebras whose GIT was analyzed by Vinberg and Levy. This leads to a classification of stable orbits in Moy-Prasad filtrations, as well as epipelagic supercuspidal representations, in terms of Z-regular elliptic automorphisms σ of the absolute root system of G. Transferred to the L-group of G, σ generates the image of tame inertia under the corresponding (wild) Langlands parameter. For unramified groups and sufficiently large p we also classify the Moy-Prasad filtrations which have semi-stable orbits, which solves the long-outstanding problem of classifying non-degenerate K-types for G(k).

Epipelagic representations and invariant theory

We classify the automorphic representations (over number fields) and the irreducible admissible representations (over local fields) of unitary groups which are not quasi-split, under the assumption that the same is known for quasi-split unitary groups. The classification of automorphic representations is given in terms of automorphic representations of general linear groups. The classification of irreducible admissible representations is given in terms of Langlands parameters.

Endoscopic Classification of Representations: Inner Forms of Unitary Groups

We dene a new cohomology set H 1 (u ! W;Z ! G) for an ane algebraic group G and a nite central subgroup, both dened over a local eld of characteristic zero, which is an enlargement of the usual rst Galois cohomology set of G. We show how this set can be used to give a precise conjectural description of the internal structure and endoscopic transfer of tempered L-packets for arbitrary connected reductive groups that extends the well-known conjectural description for quasi-split groups. In the case of real groups, we show that this description is correct using Shelstad’s work.

Rigid inner forms of real and $p$-adic groups

We provide an explicit construction of the local Langlands correspondence for general tamely-ramified reductive p-adic groups and a class of wildly ramified Langlands parameters. Furthermore, we verify that our construction satisfies many expected properties of such a correspondence. More precisely, we show that each $$L$$
 -packet we construct admits a parameterization in terms of the Langlands dual group, contains a unique generic element for a fixed Whittaker datum, satisfies the formal degree conjecture, is compatible with central and cocentral characters, provides a stable virtual character, and satisfies the expected endoscopic character identities. Moreover, we show that in the case of $$\mathrm{{GL}}_n$$
 , our construction coincides with the established local Langlands correspondence. Our techniques provide a general approach to the construction of the local Langlands correspondence for tamely-ramified groups and regular supercuspidal parameters.

Epipelagic L-packets and rectifying characters

We prove the recent conjectures of Adams-Vogan and D. Prasad on the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. The proof holds for tempered representations of quasi-split real K-groups and quasi-split p-adic classical groups (in the sense of Arthur). We also prove a formula for the behavior of the local Langlands correspondence for these groups with respect to changes of the Whittaker data.

https://msp.org/ant/2013/7-10/ant-v7-n10-p03-p.pdf

Genericity and contragredience in the local Langlands correspondence

We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups.

Tasho Kaletha

Papers

Endoscopic Classification of Representations: Inner Forms of Unitary Groups

Rigid inner forms of real and $p$-adic groups

Epipelagic L-packets and rectifying characters

Genericity and contragredience in the local Langlands correspondence

Regular supercuspidal representations